{"id":1661,"date":"2022-04-10T18:12:11","date_gmt":"2022-04-10T10:12:11","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1661"},"modified":"2025-02-26T11:34:53","modified_gmt":"2025-02-26T03:34:53","slug":"nulhomotopy_lemma","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/04\/10\/nulhomotopy_lemma\/","title":{"rendered":"\u96f6\u4f26\u5f15\u7406"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u4eca\u5929\u662f\u5446\u5728\u76db\u5e90\u5c0f\u533a\u7684\u6700\u540e\u4e00\u4e2a\u5468\u65e5, \u4e0b\u5468\u516d\u5c31\u8981\u642c\u5bb6\u4e86. \u5bf9\u65b0\u7684\u73af\u5883\u59cb\u7ec8\u6709\u4e00\u4e1d\u9690\u9690\u7684\u4e0d\u5b89, \u5305\u62ec\u4e0a\u73ed\u51fa\u884c\u65b9\u5f0f\u7684\u53d8\u5316\u7b49\u7b49. \u4f46\u4eba\u561b, \u59cb\u7ec8\u8981\u52c7\u6562\u8d70\u51fa\u81ea\u5df1\u7684\u8212\u9002\u533a\u4e0d\u662f\u5417, \u642c\u5bb6\u5982\u662f, \u5176\u5b83\u4e8b\u60c5\u4ea6\u5982\u662f. \u8fd9\u5468\u5176\u5b9e\u5bf9\u4e8e\u4ee3\u6570\u62d3\u6251\u7684\u5b66\u4e60\u8fdb\u5ea6\u582a\u5fe7, \u4e00\u65b9\u9762\u662f\u51c6\u5907\u642c\u5bb6, \u53e6\u4e00\u65b9\u9762\u662f\u5728\u738b\u8005\u5ce1\u8c37\u4e2d\u51b2\u5206\u8363\u8000\u738b\u80052333\u2026\u2026 \u73b0\u5728\u76ee\u6807\u5df2\u7ecf\u8fbe\u5230\u4e86, \u5e0c\u671b\u4e0b\u4e00\u5468\u5f00\u59cb\u80fd\u5728\u51c6\u5907\u642c\u5bb6\u4e4b\u4f59\u5c3d\u53ef\u80fd\u5730\u5b66\u4e60\u53ed~ \u672c\u6587\u4e3b\u8981\u662f\u8bb0\u5f55\u4e86\u96f6\u4f26\u5f15\u7406\u53ca\u5176\u8bc1\u660e\u8981\u70b9.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/04\/Jordan\u66f2\u7ebf\u5b9a\u7406\u8bfb\u4e66\u7b14\u8bb0.pdf\">Jordan\u66f2\u7ebf\u5b9a\u7406\u8bfb\u4e66\u7b14\u8bb0<\/a><\/p>\n<p><strong>\u96f6\u4f26\u5f15\u7406<\/strong> \u8bbe$A$\u7d27\u81f4, $f: A \\to S^2 \\backslash \\{ p, q \\}$\u8fde\u7eed, \u5e76\u4e14$p, q$\u5728$S^2 \\backslash A$\u7684\u540c\u4e00\u4e2a\u9053\u8def\u5206\u652f\u5185, \u5219$f$\u96f6\u4f26.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> <em>\u6b64\u5904\u8bc1\u660e\u8fc7\u7a0b\u53c2\u8003\u4e86\u4e0a\u8ff0\u53c2\u8003\u6587\u732e\u4e2d\u7684\u8bc1\u660e, \u5176\u4e0e\u4e66\u4e0a\u7684\u8bc1\u660e\u601d\u8def\u57fa\u672c\u4e0a\u662f\u4e00\u81f4\u7684, \u4f46\u79c1\u4ee5\u4e3a\u4e66\u4e0a\u7684\u8bba\u8ff0\u8bed\u8a00\u6709\u70b9\u95ee\u9898, \u4e0d\u5bb9\u6613\u7406\u89e3.<\/em><br \/>\n$\\\\$ \u53d6$\\rho: S^2 \\backslash \\{ q \\} \\to E^2$\u662f\u540c\u80da(\u7c7b\u4f3c\u4e8e\u7403\u6781\u6295\u5f71, \u76f8\u5f53\u4e8e\u628a$q$\u53d8\u5230\u65e0\u7a77\u8fdc\u5904), \u4e14$\\rho(p) = (0, 0)$, \u5219$f: A \\to S^2 $$ \\backslash \\{ p, q \\}$\u96f6\u4f26\u7b49\u4ef7\u4e8e$\\tilde{f} := $$ \\rho \\circ f: A \\to E^2 \\backslash $$ \\{ (0, 0) \\}$\u96f6\u4f26. \u6240\u4ee5\u53ea\u9700\u8bc1\u82e5$\\tilde{f}: A \\to $$ E^2 \\backslash \\{ (0, $$ 0) \\}$\u8fde\u7eed, \u4e14$(0, 0)$\u5728$E^2 \\backslash \\tilde{f}( $$ A)$\u7684\u65e0\u754c\u9053\u8def\u5206\u652f(\u540c\u80da\u53d8\u6362\u4fdd\u8bc1\u4e86$p$, $q$\u7684\u50cf\u70b9\u4f9d\u65e7\u5904\u4e8e\u540c\u4e00\u4e2a\u9053\u8def\u5206\u652f\u5185), \u5219$\\tilde{f}$\u662f\u96f6\u4f26\u7684.<br \/>\n$\\\\$ \u56e0\u4e3a$A$\u7d27, \u6240\u4ee5$\\tilde{f}(A)$\u7d27, \u53d6$R < 0$\u4f7f\u5f97$\\tilde{f}(A) \\subset B_R((0, 0)) =: B$. \u8bbe$x_0 \\in $$ E^2$, $x_0 \\ne B$, \u5219$x_0$\u4ea6\u5c5e\u4e8e$E^2 \\backslash \\tilde{f}(A)$\u7684\u65e0\u754c\u9053\u8def\u5206\u652f.\n$\\\\$ \u7531\u4e8e$(0, 0)$\u4e0e$x_0$\u5747\u5904\u4e8e\u65e0\u754c\u9053\u8def\u5206\u652f\u4e2d, \u6545\u5b58\u5728$E^2 \\backslash \\tilde{f}(A)$\u4e2d\u7684\u9053\u8def$\\tilde{a}$\u8fde\u63a5$(0, 0)$\u4e0e$x_0$, \u4ece\u800c\u53ef\u4ee5\u5b9a\u4e49\u540c\u4f26:$$g_t: A \\to E^2 \\backslash \\{ (0, 0) \\}, g_t(x) = \\tilde{f}(x) - \\tilde{a}(t),$$\u6ce8\u610f\u5230$g_t(x) \\in E^2 \\backslash \\{ (0, 0) \\}$, \u56e0\u4e3a$\\tilde{f}(x) \\in \\tilde{f}(A)$, $\\tilde{a}(t) \\notin \\tilde{f}(A)$. \u5e76\u4e14\u4efb\u53d6$t $$ \\in [0, 1]$, $g_t \\simeq g_0 = \\tilde{f}$, $g_1(x) = \\tilde{f}(x) - x_0$.\n$\\\\$ \u518d\u5b9a\u4e49\u540c\u4f26$$h_t: A \\to E^2 \\backslash \\{ (0, 0) \\}, h_t(x) = t\\tilde{f}(x) - x_0.$$\u56e0\u4e3a$\\left \\| t\\tilde{f}(x) \\right \\| \\le \\left \\| \\tilde{f}(x) \\right \\| < R < \\left \\| x_0 \\right \\|$, \u6240\u4ee5$h_t(x) \\ne (0, 0)$, \u4e14$h_0(x) = - $$ x_0$, $h_1(x) = \\tilde{f}(x) - x_0$, \u6545$\\tilde{f}(x) - x_0$\u96f6\u4f26, \u4ece\u800c$\\tilde{f}$\u96f6\u4f26, \u547d\u9898\u5f97\u8bc1.\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4eca\u5929\u662f\u5446\u5728\u76db\u5e90\u5c0f\u533a\u7684\u6700\u540e\u4e00\u4e2a\u5468\u65e5, \u4e0b\u5468\u516d\u5c31\u8981\u642c\u5bb6\u4e86. \u5bf9\u65b0\u7684\u73af\u5883\u59cb\u7ec8\u6709\u4e00\u4e1d\u9690\u9690\u7684\u4e0d\u5b89, \u5305\u62ec\u4e0a\u73ed\u51fa\u884c\u65b9\u5f0f\u7684\u53d8 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/04\/10\/nulhomotopy_lemma\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u96f6\u4f26\u5f15\u7406<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1661"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1661"}],"version-history":[{"count":32,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1661\/revisions"}],"predecessor-version":[{"id":3647,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1661\/revisions\/3647"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1661"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1661"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1661"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}